Wapperdude:
Thanks for taking a look. Turns out V2013 and my old V2003 behave the same way as your V2007, so it had to be something deeper. And based on your input, I was able to find out exactly what it was, and subsequently achieve my goal of a Visio-connected "underlined box".
What my original code did was to marry the box and line together irrespective of any connection, and the two shapes just traveled together according to the line's endpoint formulae. Without your suggested GlueTo, there was no connection in the Visio sense, and therefore no Connects.Count.
Taking it a step further, the final code to connect the two turned out to be even simpler than your approach...
Set TheLineConnectionsX1 = TheLine.Cells("Connections.X1")
Set TheBoxConnectionsX1 = TheBox.Cells("Connections.X1")
TheLineConnectionsX1.GlueTo TheBoxConnectionsX1
...which glues the two connection points to each other in the Visio sense, yielding a Connects.Count of 2, as expected, and an underlined box, as desired.
Thanks again for putting me on the path to solving my problem. You're already making it a Happy New Year, even though it's only a few dozen hours old.
- Ken